And if you have a shape of known cross section, the integral becomes much simpler.

Like this paraboloid, for example. The equation is z = -(x² + y²) + 10. The cross section is a circle of radius √[-z + 10] = x² + y². The circumference is then 2π√[-z + 10] = x² + y².

You then integrate the circumference multiplied by the differential of z, dz.

In coming up with this example, I've realized a deficiency in my instruction that I'm sure will be filled at a later date, but I want to know now. How would I set up said integral? Always before when I've done problems like this, they've given me a nice neat expression for r. Now...I don't know.